Tagged Questions
1
vote
1answer
30 views
Two questions on $\limsup$: do nested ones commute and sending $n$ to $-\infty$
As for the second question, this is really just me wondering on definitions. For a function $f:\mathbb{R}\to\mathbb{R}$ define
$$\limsup_{x\to+\infty}f(x):=\lim_{x\to+\infty}\sup_{z\geq x}f(x).$$
...
1
vote
3answers
110 views
For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function?
Alternatively, how does the definition of a limit guarantee that if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
3
votes
0answers
44 views
Understanding the roots of homomorphism and homeomorphism
I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
2
votes
3answers
109 views
Example of a (dis)continuous function
The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous?
I ...
5
votes
1answer
89 views
Is uncountably summation defined?
We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$
Is it defined?
1
vote
1answer
50 views
Limit point definition
I have read the definition of a limit point of a set in Real Analysis.
The definition goes like:
A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
1
vote
2answers
73 views
What is the meaning of the expression $\liminf f_n$?
I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
1
vote
4answers
95 views
Definition of a metric
I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
1
vote
3answers
75 views
Difference between closure and the boundary
I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the ...
0
votes
0answers
51 views
Improper or Undefined
Let $f(x)=0 $ if $x\neq 1 $ and $f(1)=\infty $ then the Riemann integral
$\int_{0} ^1 f(x)$ $ dx $ = $ 0 $ or is it undefined?
If we take it as a legitimate function for improper Riemann ...
10
votes
6answers
347 views
How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?
I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...
3
votes
2answers
110 views
Motivation for a new definition of the derivative using the concept of average velocity
As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if
$$
\lim_{x \to a} ...
1
vote
4answers
127 views
If capital letters are supposed to be sets, why is $N$ used as a number?
If capital letters are supposed to be sets, why is $N$ used as a number? For example, in this definition of a Cauchy Sequence it says:
A sequence ${p_n}$ in a metric space $X$ is said to be a ...
1
vote
1answer
32 views
A Quadratic Maximum?
What does the following mean?
Context: Laplace integrals
Consider the integral $$\int_a^b f(t) \exp(x\phi(t))\,\,\,dt$$ where $\phi(t) $ achieves its maximum at some $t=c\in (a,b)$. Assume that ...
1
vote
1answer
74 views
Definition for series with negative index and order of taking limits
I have thee questions and they seem all related to me and every number i say is complex number below.
My definition for $\sum_{k=0}^n=a_k$ is by induction. That is, given finite set of numbers ...
5
votes
2answers
198 views
Why does the condition of a function being differentiable always require an open domain?
Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset $A$ of $\mathbb{R}^n$ to $\mathbb{R}^m$, $f$ is said to be differentiable if it can be ...
5
votes
4answers
277 views
Can open sets be an open cover, for itself?
I have Baby Rudin's book with me and it clearly defines a cover to be open. In a followup, it defines a set $K$ to be compact if every open cover of $K$ contains a finite subcover.
And the rest I ...
1
vote
0answers
151 views
different kind of convergence in Real analysis
Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
0
votes
0answers
54 views
What is derived number definition? ( in Vitaly covering)
In Vitali covering definition i see "derived number" word, but I dont know what that mean.
Example for vitali covering:
If $f$ is strictly increasing and
$$E=\{x: \text{ there is a derived number } ...
7
votes
2answers
195 views
True Definition of the Real Numbers
I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
6
votes
1answer
73 views
Can we give a definition of the cotangent based on a functional equation?
I've recently learned that the cotangent satisfies the following functional equation:
$$\dfrac1{f(z)}=f(z)-2f(2z)$$
(true for $f(z)\neq 0$).
Can we solve this equation for real or complex ...
1
vote
1answer
280 views
Bounded Set: definition
I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds.
Now i have a quiz in which I must choose the ...
1
vote
2answers
72 views
visualisation of pointwise boundedness
A sequence of continuous functions $(f_n\colon[a,b]\to\mathbf{R})_{n}$ is said to be point-wise bounded if for all $x\in[a,b]$ there is a $R_x>0$ such that $$|f_n(x)|\le R_x\quad\mbox{for all }n.$$
...
8
votes
5answers
391 views
Proving that $a + b = b + a$ for all $a,b \in\mathbb{R}$
Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting ...
2
votes
2answers
328 views
Local definition of Hölder continuity
What does it mean for a continuous function $ f $ on $ \mathbb{R} $ to be Hölder continuous with exponent $ \alpha $ at a point $ x_0 $ ?
I only now the global definition:
A function $ f $ on $ ...
2
votes
2answers
263 views
What are the primitive notions of real analysis?
My dad introduced my to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition?
...
1
vote
3answers
730 views
why do we use 'non-increasing' instead of decreasing?
In english based math language it seems that
non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing)
decreasing $\Longleftrightarrow$ strict less ( strict decreasing)
...
0
votes
2answers
301 views
How many 'supremum(s)' and 'infimum(s)' can a set have?
I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum ...
0
votes
1answer
183 views
Why is this definition of an additive inverse significant
In the process of learning Real Analysis, I encountered a definition of an additive inverse of a cut $\alpha$ to be
$$\text {add inv of } \alpha \colon= \{p:\exists r>0 \text{ s.t.} (-p-r)\notin ...
2
votes
0answers
194 views
How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?
While reading this post, I stumbled across these definitions (Wiki_german)
$$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$
and
$$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$
The last one seems ...
17
votes
6answers
1k views
Why does the Dedekind Cut work well enough to define the Reals?
I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals.
I just can't get the ...
4
votes
5answers
469 views
What is a Real Number?
I believe I'm over thinking it, but I want to be 100% sure.
A Real Number is any number, correct? Whether it be an integer or something else.
It's the set $\mathbb R$ from $(-\infty, +\infty)$ ...
2
votes
0answers
183 views
why are curl and divergence defined the way they are?
Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
1
vote
1answer
138 views
Rephrasing Munkres' Theorem Re: Inverses of Jacobians
Below are theorems from Munkres' "Analysis on Manifolds". The proof of Theorem 7.4 on the right invokes the chain rule, stated on the left. The conditions of Theorem are somewhat strange and appear ...
5
votes
1answer
128 views
What motivates discrepancies between the definitions of “continuous” and “limit”?
I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the ...
5
votes
3answers
498 views
Why not define 'limits' to include isolated points?
If I understand correctly, most definitions of 'limits' require that the function either a) be defined in an open neighborhood around the relevant point or b) more permissively, that the relevant ...
9
votes
3answers
474 views
Differentiable at a point
My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
8
votes
2answers
360 views
Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials
[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below]
in my attempt to find a precise version of the 'definitions' usually given when ...
1
vote
1answer
114 views
Field of sets versus a field as an algebraic structure
During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
